A Glorious Game

Here’s a classic brain teaser: “At the close of last season, of the footballers of my acquaintance four had broken their left arm, five had broken their right arm, two had the right arm sound, and three had sound left arms.”

Can you discover from that statement what is the smallest number of players that the speaker could be acquainted with?

first you start by “minimal” number of players, you have 4 groups:
1) with broken right arm–> 5 (A)
2) with broken left arm –> 4 (B)
3) with right arm sound–> 2 (C)
4) with left arm sound–> 3 (D)
note: the letters represent each group

the “minimal” number of players is met when every player can be in as many groups he can be so, in conjunts is where the more unions exists So:
1)the 2 of C exist in A,B,D–>so A(5-2=3) B(4-2=2) C(2-2=0) D(3-2=1)
2)the 1 left in D exist in A,B–> so A (3-1=2) B(2-1=1)D(1-1=0)
3)the 1 left in B exist in A–> so A( 2-1=1)B(1-1=0)
4)The 1 left in A exist only in A(1-1=0)
so the minimal number is 2+1+1+1= 5
and the categorization is this:
1) 2 have broken left and right arm and also have sound right and left arm
2) 1 have left and right arm broken but only left arm sound
3) 1 have left and right arm borken but neither arm sound
4) 1 have only the right arm broken

Fewest number of players would be 7. The number of broken left arms or broken right arms are not mutually exclusive. There could be 4 players with both arms broken, 1 with their right arm broken and 2 with neither arm broken. 4+2+1=7

I used to believe that sound arm was something refering a luxation or a dislocation, thats why i thought that sound arm and broken arm were not mutually exclusive, but cool puzzle indded, i learn that sound arm means something close to healthy arm, and my answer in that case will be:
1) 4 with both arms broken
2) 1 with the right arm broken and the left ok (sound)
3) 2 with the two arms ok (sound)
4+1+2=7 the same answer that everyone post it above but me

My answer is 5 players. Using the Venn diagram, you can easily figure it out. I agree with @suineg. You have to have in your mind that players can sound and also break both of their arms. You will need to make 3 Venn diagrams

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